Can you be a doctor without a state license?

Yes: Once you graduate from medical school you are a doctor. However, you must go through residency and obtain a state license in order to practice as a physician. You can not practice as a physician without a license.

Not a practicing one: To practice medicine, you must be licensed in the state. Or, you could be practicing as a resident in a training program, under a medical training license associated with the hospital you're working in. It's possible to graduate from medical school with your md or do degree and never become licensed or practice. Technically one would be a "doctor" but could not practice legally.